Differential operators $D_x$, $D_y$, and $D_z$ are formed using the
action of the $3$-dimensional discrete Heisenberg group $G$ on a set
$S$, and the operators will act on functions on $S$. The Laplacian
operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with
variable differences which can be associated to a unitary
representation of $G$ on the Hilbert space $L^2(S)$. Using techniques
from harmonic analysis and representation theory, we show that the
Laplacian operator is locally solvable.